Existence results of positive solutions for nonlinear cooperative elliptic systems involving fractional Laplacian

TL;DR

This paper establishes the existence of positive solutions for a class of nonlinear elliptic systems involving fractional Laplacians and gradient terms, using topological methods and classical scaling techniques.

Contribution

It provides new existence results for positive solutions of fractional elliptic systems with gradient dependence, extending previous work to more general nonlinearities.

Findings

Existence of at least one positive solution under certain conditions.

Application of classical scaling method and topological degree theory.

Results applicable to smooth bounded domains in Euclidean space.

Abstract

In this article, we prove existence results of positive solutions for the following nonlinear elliptic problem with gradient terms: \begin{eqnarray*} \left\{\begin{array}{l@{\quad }l} (-\Delta)^\alpha u=f(x,u,v,\nabla u, \nabla v) &{\rm in}\,\,\Omega,\\ (-\Delta)^\alpha v=g(x,u,v,\nabla u, \nabla v) &{\rm in}\,\,\Omega,\\ u=v=0\,\,&{\rm in}\,\,\R^N\setminus\Omega, \end{array} \right. \end{eqnarray*} where $(-\Delta)^\alpha$ denotes the fractional Laplacian and $ \Omega $ is a smooth bounded domain in $ \R^N $. It shown that under some assumptions on $ f $ and $ g $, the problem has at least one positive solution $(u,v)$. Our proof is based on the classical scaling method of Gidas and Spruck and topological degree theory.